3.11 \(\int (-1-\cot ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=35 \[ \frac{1}{2} \cot (x) \sqrt{-\csc ^2(x)}-\frac{1}{2} \tan ^{-1}\left (\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}\right ) \]

[Out]

-ArcTan[Cot[x]/Sqrt[-Csc[x]^2]]/2 + (Cot[x]*Sqrt[-Csc[x]^2])/2

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Rubi [A]  time = 0.0255565, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {3657, 4122, 195, 217, 203} \[ \frac{1}{2} \cot (x) \sqrt{-\csc ^2(x)}-\frac{1}{2} \tan ^{-1}\left (\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-1 - Cot[x]^2)^(3/2),x]

[Out]

-ArcTan[Cot[x]/Sqrt[-Csc[x]^2]]/2 + (Cot[x]*Sqrt[-Csc[x]^2])/2

Rule 3657

Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]

Rule 4122

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \left (-1-\cot ^2(x)\right )^{3/2} \, dx &=\int \left (-\csc ^2(x)\right )^{3/2} \, dx\\ &=\operatorname{Subst}\left (\int \sqrt{-1-x^2} \, dx,x,\cot (x)\right )\\ &=\frac{1}{2} \cot (x) \sqrt{-\csc ^2(x)}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x^2}} \, dx,x,\cot (x)\right )\\ &=\frac{1}{2} \cot (x) \sqrt{-\csc ^2(x)}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}\right )\\ &=-\frac{1}{2} \tan ^{-1}\left (\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}\right )+\frac{1}{2} \cot (x) \sqrt{-\csc ^2(x)}\\ \end{align*}

Mathematica [A]  time = 0.0796303, size = 48, normalized size = 1.37 \[ -\frac{\csc \left (\frac{x}{2}\right ) \sec \left (\frac{x}{2}\right ) \left (-\log \left (\sin \left (\frac{x}{2}\right )\right )+\log \left (\cos \left (\frac{x}{2}\right )\right )+\cot (x) \csc (x)\right )}{4 \sqrt{-\csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 - Cot[x]^2)^(3/2),x]

[Out]

-(Csc[x/2]*(Cot[x]*Csc[x] + Log[Cos[x/2]] - Log[Sin[x/2]])*Sec[x/2])/(4*Sqrt[-Csc[x]^2])

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Maple [A]  time = 0.016, size = 32, normalized size = 0.9 \begin{align*}{\frac{\cot \left ( x \right ) }{2}\sqrt{-1- \left ( \cot \left ( x \right ) \right ) ^{2}}}-{\frac{1}{2}\arctan \left ({\cot \left ( x \right ){\frac{1}{\sqrt{-1- \left ( \cot \left ( x \right ) \right ) ^{2}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1-cot(x)^2)^(3/2),x)

[Out]

1/2*cot(x)*(-1-cot(x)^2)^(1/2)-1/2*arctan(cot(x)/(-1-cot(x)^2)^(1/2))

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Maxima [B]  time = 1.6474, size = 383, normalized size = 10.94 \begin{align*} \frac{{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) -{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right ) + 2 \,{\left (\sin \left (3 \, x\right ) + \sin \left (x\right )\right )} \cos \left (4 \, x\right ) - 2 \,{\left (\cos \left (3 \, x\right ) + \cos \left (x\right )\right )} \sin \left (4 \, x\right ) - 2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \sin \left (3 \, x\right ) + 4 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \cos \left (x\right ) \sin \left (2 \, x\right ) - 4 \, \cos \left (2 \, x\right ) \sin \left (x\right ) + 2 \, \sin \left (x\right )}{2 \,{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1-cot(x)^2)^(3/2),x, algorithm="maxima")

[Out]

1/2*((2*(2*cos(2*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 - sin(4*x)^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^
2 + 4*cos(2*x) - 1)*arctan2(sin(x), cos(x) + 1) - (2*(2*cos(2*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 - s
in(4*x)^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^2 + 4*cos(2*x) - 1)*arctan2(sin(x), cos(x) - 1) + 2*(sin(3*x) + s
in(x))*cos(4*x) - 2*(cos(3*x) + cos(x))*sin(4*x) - 2*(2*cos(2*x) - 1)*sin(3*x) + 4*cos(3*x)*sin(2*x) + 4*cos(x
)*sin(2*x) - 4*cos(2*x)*sin(x) + 2*sin(x))/(2*(2*cos(2*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 - sin(4*x)
^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^2 + 4*cos(2*x) - 1)

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Fricas [C]  time = 1.7396, size = 232, normalized size = 6.63 \begin{align*} \frac{{\left (-i \, e^{\left (4 i \, x\right )} + 2 i \, e^{\left (2 i \, x\right )} - i\right )} \log \left (e^{\left (i \, x\right )} + 1\right ) +{\left (i \, e^{\left (4 i \, x\right )} - 2 i \, e^{\left (2 i \, x\right )} + i\right )} \log \left (e^{\left (i \, x\right )} - 1\right ) + 2 i \, e^{\left (3 i \, x\right )} + 2 i \, e^{\left (i \, x\right )}}{2 \,{\left (e^{\left (4 i \, x\right )} - 2 \, e^{\left (2 i \, x\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1-cot(x)^2)^(3/2),x, algorithm="fricas")

[Out]

1/2*((-I*e^(4*I*x) + 2*I*e^(2*I*x) - I)*log(e^(I*x) + 1) + (I*e^(4*I*x) - 2*I*e^(2*I*x) + I)*log(e^(I*x) - 1)
+ 2*I*e^(3*I*x) + 2*I*e^(I*x))/(e^(4*I*x) - 2*e^(2*I*x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- \cot ^{2}{\left (x \right )} - 1\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1-cot(x)**2)**(3/2),x)

[Out]

Integral((-cot(x)**2 - 1)**(3/2), x)

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Giac [C]  time = 1.15838, size = 46, normalized size = 1.31 \begin{align*} -\frac{1}{4} \,{\left (\frac{2 i \, \cos \left (x\right )}{\cos \left (x\right )^{2} - 1} - i \, \log \left (\cos \left (x\right ) + 1\right ) + i \, \log \left (-\cos \left (x\right ) + 1\right )\right )} \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1-cot(x)^2)^(3/2),x, algorithm="giac")

[Out]

-1/4*(2*I*cos(x)/(cos(x)^2 - 1) - I*log(cos(x) + 1) + I*log(-cos(x) + 1))*sgn(sin(x))